How are hydrodynamics and field theories related? If you’d asked me early last week I wouldn’t have much to say. However, after hearing Pavel Kovtun’s talk at the Caltech Physics Colloquium last Thursday, I now see that they’re pretty deeply related.

I’ve always seen hydrodynamics as equivalent to fluid mechanics, which I dismissed as “an engineering subject.” Also, the math gets ugly fast! As for field theories, I immediately think quantum field theory, where the math is also tough but the concepts are more interesting to me. Kovtun illuminated the connection between the two with a deceptively simple statement:

So if all conservation laws can be written in the form of this continuity equation, why isn’t it more familiar to me?

Well, after some searching, it became clear that the main reason this isn’t easily recognized outside of fluids and electrodynamics (Ampere’s Law with Gauss’s Law) is because the density and flux quantities don’t always have a simple physical interpretation. I was hoping that there’d be some magical version for conservation of energy, or Einstein’s famous energy-mass relation, or the simple classical conservation of linear momentum. The closest I found was Gauss’s law for gravity, which becomes particularly interesting when discussing dark matter, but we can dive into that in a future post. For now, suffice it to say I’m still on the search for other interesting continuity equations.